| L(s) = 1 | + (−0.732 + 2.73i)2-s + (−7.91 − 4.56i)3-s + (−6.92 − 4i)4-s + (9.46 + 35.3i)5-s + (18.2 − 18.2i)6-s + (−10.9 + 18.9i)7-s + (16 − 15.9i)8-s + (1.22 + 2.11i)9-s − 103.·10-s − 135. i·11-s + (36.5 + 63.2i)12-s + (−72.3 − 270. i)13-s + (−43.8 − 43.8i)14-s + (86.4 − 322. i)15-s + (31.9 + 55.4i)16-s + (−450. − 120. i)17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.683i)2-s + (−0.879 − 0.507i)3-s + (−0.433 − 0.250i)4-s + (0.378 + 1.41i)5-s + (0.507 − 0.507i)6-s + (−0.223 + 0.387i)7-s + (0.250 − 0.249i)8-s + (0.0151 + 0.0261i)9-s − 1.03·10-s − 1.11i·11-s + (0.253 + 0.439i)12-s + (−0.428 − 1.59i)13-s + (−0.223 − 0.223i)14-s + (0.384 − 1.43i)15-s + (0.124 + 0.216i)16-s + (−1.55 − 0.417i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0777 + 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0777 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.250230 - 0.270514i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250230 - 0.270514i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.732 - 2.73i)T \) |
| 37 | \( 1 + (1.35e3 - 207. i)T \) |
| good | 3 | \( 1 + (7.91 + 4.56i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-9.46 - 35.3i)T + (-541. + 312.5i)T^{2} \) |
| 7 | \( 1 + (10.9 - 18.9i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 135. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (72.3 + 270. i)T + (-2.47e4 + 1.42e4i)T^{2} \) |
| 17 | \( 1 + (450. + 120. i)T + (7.23e4 + 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-103. - 387. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-716. + 716. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (380. + 380. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (275. + 275. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 + (1.38e3 + 801. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.66e3 + 1.66e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + 1.22e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (-470. - 815. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (2.56e3 + 687. i)T + (1.04e7 + 6.05e6i)T^{2} \) |
| 61 | \( 1 + (4.64e3 - 1.24e3i)T + (1.19e7 - 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-1.07e3 - 620. i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (3.39e3 - 5.87e3i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 2.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (105. + 393. i)T + (-3.37e7 + 1.94e7i)T^{2} \) |
| 83 | \( 1 + (4.50e3 + 7.80e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (1.40e3 - 5.26e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (4.43e3 - 4.43e3i)T - 8.85e7iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64918385755937364176979553446, −12.52056110390990329813370357921, −11.10180303449713087037689211860, −10.41742713769472778840466725088, −8.805500097457931194159780328620, −7.26280236121001458551771402075, −6.31914129297292175372852884143, −5.55259016279891785830159955176, −2.96854431723699738451000425618, −0.22215343343641197900397625806,
1.71402047932870541738419165969, 4.46316003090124041531641363088, 5.02901885863675167378731105432, 6.97271091908230771857076399138, 8.994978806127687043094918652746, 9.526213270194846778869956779840, 10.91830391563713906065983577437, 11.77778635944606707431405849903, 12.88349611426342538653011683762, 13.65406543640266948553197378563
